Improved distance queries in planar graphs

TL;DR

This paper introduces three new data structures for efficient distance queries in planar graphs, optimizing for different tradeoffs between preprocessing time, storage space, and query time.

Contribution

It presents three novel data structures that improve query time, preprocessing time, or space efficiency for distance queries in planar graphs, leveraging advanced decomposition and Monge properties.

Findings

Improved query time for linear space data structures.

Faster preprocessing for space-bound data structures.

Enhanced query efficiency with longer preprocessing times.

Abstract

There are several known data structures that answer distance queries between two arbitrary vertices in a planar graph. The tradeoff is among preprocessing time, storage space and query time. In this paper we present three data structures that answer such queries, each with its own advantage over previous data structures. The first one improves the query time of data structures of linear space. The second improves the preprocessing time of data structures with a space bound of O(n^(4/3)) or higher while matching the best known query time. The third data structure improves the query time for a similar range of space bounds, at the expense of a longer preprocessing time. The techniques that we use include modifying the parameters of planar graph decompositions, combining the different advantages of existing data structures, and using the Monge property for finding minimum elements of matrices.